A day after the Senate passed its immigration bill, RNC Chair Reince Priebus claimed there was “broad consensus” in the Republican party for “comprehensive immigration reform.”

“We need comprehensive immigration reform,” Priebus told CNN’s Peter Hamby on Friday. “I don’t think we can continue to drift along with this mess of immigration laws that we have. And a mess that in many regards has been the results of our government not even enforcing the laws that are in place. There is plenty of blame to go around for why we are in this position, but I think it’s about time that we address it.”

He then asserted it was “clear that there is pretty broad consensus in the party, in the Republican Party, that we need comprehensive immigration reform” and said he knew the House leadership is “committed” to drafting a comprehensive bill.

Republican House leaders have indicated that they will not take up the Senate’s bill, but have signalled they will draft “comprehensive” legislation they hope to pass in order to go to conference with the Senate. Conservatives, though, would like a narrower approach that focuses primarily on securing–and funding–border security first.

Priebus’s assertion that there is a “broad consensus” within the party for reform will surely be tested as the House takes up its own version of the bill.

Already, influential conservatives like Sarah Palin have warned the House not to betray America’s working class of all ethnicities. Palin even suggested conservaties may leave the Republican party if the GOP-controlled House passes “comprehensive” immigration reform that lowers the wages of Americans, raises unemployment, and does little to improve border security.

The Senate passed its immigration legislation 68-32 on Thursday, with 14 Republicans voting for the bill. The fight now moves to the House.